where x = tan u and u = arctan x
I can't see where acrtan has come from. What part of the integrand is arctan?
They chose x= tan(u) in order to use the trig identity . There is NO part of the integral that is arctan. You need u= arctan x when you want to convert the completed integral back to x.
The point is that if you let x= tan(u) then while so your integral becomes
. You can integrate that with the trig identity
.